For the second model (hereafter M2), we combine the results from
weak lensing and 2PCCF to constrain the halo mass.
Different from M1, here we use the MCMC to fit the lensing results of
cAGN, cSF\(^{\rm c}\)  and cQ\(^{\rm c}\)  simultaneously, and use the
ratios of the 2PCCFs at large scales between the three samples
as additional constraints. To this end, we use the halo mass
estimated at each MCMC chain step to calculate the halo bias from
the analytical formula given in [1]}. We then obtain
the model bias ratios, cAGN\(/\) cSF\(^{\rm c}\)  and cAGN\(/\) cQ\(^{\rm c}\) ,
and fit them to the corresponding ratios obtained from the observed 2PCCF.
The likelihood function for the bias term is similar to
Equation (REF ), except that the
covariance matrix \(C_{2}\)  is built from bootstrap sampling,

(R) = -12(Rwp-Rhb)TC2-1(Rwp-Rhb),


where \(R_{\rm hb}\)  and \(R_{w_{\rm p}}\)  are the model bias ratio
and the 2PCCF ratio between AGNs and the corresponding
control sample, respectively. We only use the 2PCCF ratios
on large-scales in the fitting: \(r_{\rm p}>1\, h^{-1}{\rm {Mpc}}\)  for cAGN\(/\) cSF\(^{\rm c}\) 
and \(r_{\rm p}>4\, h^{-1}{\rm {Mpc}}\)  for cAGN\(/\) cQ\(^{\rm c}\) .
The reason for these choices and the robustness of the method are
described in Section REF .
